The Republican incumbent alleged he was the victim of interference from “phony polls” as well as “big media, big money and big tech” after launching legal fights to stay in office.
He called for a “little patience” and “calm”.
At a White House press conference on Thursday night, Mr Trump said: “If you count the legal votes I easily won. If you count the illegal votes, they can try to steal the election from us.”
Hours later, he tweeted that he had “easily” won and that the Supreme Court should be allowed to decide who clinches victory.
In a string of tweets, he wrote: “I easily WIN the Presidency of the United States with LEGAL VOTES CAST. The OBSERVERS were not allowed, in any way, shape, or form, to do their job and therefore, votes accepted during this period must be determined to be ILLEGAL VOTES. U.S. Supreme Court should decide!”
The presidential election remains too close to call after polls closed on Tuesday, but former vice-president Mr Biden remains the favourite after winning three key battleground states.
Pennsylvania was expecting to announce its result by the end of the day. If it goes to Mr Biden, so would the White House.
Mr Biden’s victories have him at 253 of the 270 Electoral College votes needed to win the election.
He said he had “no doubt” he will win the presidency as he urged for calm and patience and re-iterated that “each ballot must be counted”.
In a brief statement to the nation earlier in the evening, Mr Biden added that “democracy is sometimes messy, it sometimes requires a little patience”.
Mr Trump, with 214 electoral votes, faced a much higher hurdle than his rival and his campaign has engaged in a flurry of legal activity, requesting a recount in some key states.
To reach 270, he needed to claim all four remaining battlegrounds: Pennsylvania, North Carolina, Georgia and Nevada.
Mr Trump said his party thought they would win the election “very easily” but that there would be “a lot of litigation” involved.
“We have so much evidence, so much proof and it’s going to end up, perhaps, at the highest court in the land”, he said. “We think there will be a lot of litigation because we can’t have an election stolen like this.”
The president reminded reporters in the White House press briefing room that he had been saying “for months” that mail-in ballots would be a “disaster”.
Mr Biden already has won the fiercely contested prizes of Michigan and Wisconsin, part of the “blue wall” that slipped away from Democrats four years ago.
With millions of votes yet to be tabulated, Mr Biden already had received more than 71 million votes, the most in history.
“Each ballot must be counted and that’s what we’re going to see going through now and that’s how it should be,” Mr Biden said from his home town of Wilmington, Delaware.
“We continue to feel very good about where things stand, we have no doubt that when the count is finished Senator Harris and I will be declared the winners.
“So I ask everyone to stay calm, all the people to stay calm, the process is working, the count is being completed and we’ll know very soon.”
It was a stark contrast to the approach of Mr Trump, who early on Wednesday morning falsely claimed that he had won the election.
Mr Trump’s campaign has requested a recount in Wisconsin and filing lawsuits in Pennsylvania, Michigan and Georgia.
Statewide recounts in Wisconsin have historically changed the vote tally by only a few hundred votes. Mr Biden led by more than 20,000 ballots out of nearly 3.3 million counted.
During his press briefing, Mr Trump claimed the Republicans were “winning in all the key locations” but then their lead had been “whittled away in secret”.
“We were winning in all the key locations by a lot, actually, and then our numbers started miraculously getting whittled away in secret”, he said.
Referring to legally permissible observers, the US President claimed that when observers arrived, they were told to be “100 feet away” from the count.
“When the observers got there, they wanted them 60, 70 feet away, 80 feet, 100 feet away, or outside the building, to observe people inside the building”, he told reporters.
On the postal vote system, Mr Trump added: “I’ve been talking about mail-in voting for a long time. It’s really destroyed our system, it’s a corrupt system.”
It was unclear when a national winner would be determined after a long, bitter campaign dominated by the coronavirus and its effects on Americans and the national economy.